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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(990, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,45,18]))
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pari:[g,chi] = znchar(Mod(283,990))
\(\chi_{990}(7,\cdot)\)
\(\chi_{990}(13,\cdot)\)
\(\chi_{990}(193,\cdot)\)
\(\chi_{990}(277,\cdot)\)
\(\chi_{990}(283,\cdot)\)
\(\chi_{990}(337,\cdot)\)
\(\chi_{990}(403,\cdot)\)
\(\chi_{990}(457,\cdot)\)
\(\chi_{990}(547,\cdot)\)
\(\chi_{990}(607,\cdot)\)
\(\chi_{990}(673,\cdot)\)
\(\chi_{990}(733,\cdot)\)
\(\chi_{990}(787,\cdot)\)
\(\chi_{990}(853,\cdot)\)
\(\chi_{990}(877,\cdot)\)
\(\chi_{990}(943,\cdot)\)
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sage:chi.galois_orbit()
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pari:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
\((551,397,541)\) → \((e\left(\frac{1}{3}\right),-i,e\left(\frac{3}{10}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 990 }(283, a) \) |
\(1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) |
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sage:chi.jacobi_sum(n)
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sage:chi.gauss_sum(a)
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pari:znchargauss(g,chi,a)
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sage:chi.jacobi_sum(n)
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sage:chi.kloosterman_sum(a,b)
